michel 2

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Finite groups of Lie type

Steinberg [Endomorphisms of linear algebraic groups, 1968] has studiedthe situation where Gis a reductive algebraic group over an algebraicallyclosed field and Fis an algebraic endomorphism such that GF (the fixed

points) is finite. Then GF is called a finite group of Lie type.

He has classified the possibilities. IfG is simple then it is an algebraicgroup over the algebraic closure Fqof a finite field Fq, and

either F is the Frobenius attached to an Fq-structure,

orG is of type B2, F4 orG2 and F2 is a Frobenius attached to an

Fq-structure.

In the first two of the last cases qis a power of 2 and in the third case a

power of 3; the groups G

F

in these cases where discovered by Suzuki andRee.

Jean Michel (Universite Paris VII) Finite groups of Lie type Berlin, September 2009 1 / 15

What is a Frobenius?Let qbe a power of the prime p.On the affine line A1 = Spec(Fq(T)) we have the Galois action

aiTi

aqiT

i. Since in characteristic pwe have (a+b)q =aq +bq

the algebraic morphism given by T Tq, i.e.

aiTi

aiTqi has the

same fixed points as the Galois action, since the composed is justaiT

i (

aiTi)q.

It is called the Frobeniusmorphism.

In general an affine variety Spec Awhere A is an Fq-algebra has anFq-structure ifA= A0 FqFqwhere A0 is an Fq-algebra. This structure isequivalently given by the Frobenius a aq ; we recover A0 as{x |F(x) =xq}.Another morphism F is the Frobenius associated to another Fq-structureiff there exists n such that Fn =Fn.Example: On GLn(Fq) the standard Frobenius sends a matrix {aij}to{aqij}, and the fixed points are GLn(Fq).


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Lang-Steinberg theorem

Theorem (Lang-Steinberg)

If F is an algebraic endomorphism of the affine connected algebraic group

G such that GF is finite, then the map x xF(x1) is surjective.

Equivalently H1(F, G) = 1. Indeed H1(F, G) is the set ofF-conjugacyclasses ofG, where F-conjugacy ofx byy sends x to yxF(y1).

Note that it results form Steinbergs classification that any gG is stablebyFm form large enough (since it is defined over some finite field, and apower ofF is a Frobenius); it also follows that gFstill has finitely manyfixed points (since a power is a Frobenius).


CorollaryCorollary (of Lang-Steinberg)

If G is not connected we have H1(F, G) =H1(F, G/G0).

Proof:

It is clear that two F-conjugate elements ofG have F-conjugateimages in G/G0, thus we have natural surjective map

H1(F, G) H1(F, G/G0).

Conversely, ifg and g areF-conjugate modulo G0, we havehg=xgF(x1) for some h G0 and xG.Now we may apply the Lang-Steinberg theorem in G0 with themorphism gF to write h=y1gF(y)g1, whenceg=yxgF(x1y1) so g and g areF-conjugate.

Actually, ifH is a normal subgroup ofG, we have an exact sequence

1 HF GF (G/H)F H1(F, H) H1(F, G) H1(F, G/H)

so ifH is connected we have (G/H)F

=GF

/HF

.Jean Michel (Universite Paris VII) Finite groups of Lie type Berlin, September 2009 4 / 15

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Another corollary

Corollary

Suppose the connected algebraic group G acts on the algebraic variety X

in an F-equivariant way, i.e. F acts on X and F(g(x)) =F(g)(F(x)), andtransitively. Then

XF =

and the GF-orbits of G on XF are in bijection with H1(F, CG(x))where CG(x) is the stabilizer of some point xX

F.

Proof:

Given xXwe have F(x) =h(x) for some h G. IfgF(g1) =h1

then F(g1(x)) =F(g1)F(x) =F(g1)h(x) =g1(x).

We have a map from XF to H1(F, CG(x)) as follows: given yXF

there exists gGsuch that y=g(x). From F(y) =ywe getg1F(g) CG(x). We map the GF-orbit ofyto the image of

g1F(g) in H1(F, CG(x)). It is easy to check it is a well-definedbijective map.


Tori and Borel subgroups

A maximal torusis a maximal connected abelian subgroup.Example: In GLn, the diagonal matrices.A maximal torus is isomorphic to Gnm where n is the rankofG.

A Borel subgroup B is a maximal connected solvable subgroup. We haveB=U T where U is the unipotent radical ofB and T is a maximaltorus.

Example: In GLn, we may take for Bthe upper triangular matrices; thenU=upper triangular matrices with 1s on the diagonal.

All pairs TB where T is a maximal torus and Ba Borel subgroup areconjugate, thus there exists such an F-stable pair.We have NG(B) =B and NG(T) B=CG(T) =T, thusNG(TB) =Tand since T is connected H

1(F, T) = 1 and thus theF-stable pairs are GF-conjugate.


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The Weyl group and a reflection coset

The GF-orbits ofF-stable T correspond toH1(F, NG(T)) =H

1(F, NG(T)/T) =H1(F, W).

Example: In GLn, we have NG(diagonal matrices) = monomial matricesand the quotient is the symmetric group Sn.

Let X(T) := Hom(T,Gm) = Hom(Gm,Gm)n Zn since

Hom(Gm,Gm) ={x xa}aZ.Then NG(T) acts on X(T) and Tacts trivially so W acts.

W is a reflection group on Q X(T) and so also on C X(T) = Cn.

F acts on X(T) and normalizes W. We thus have a reflection coset WF.We assume now (automatic ifGsimple or equivalently W irreducible) thateither F is the Frobenius attached to an Fpa-structure or F

2 is.

In the first case we set q=pa

and in the second case q=pa/2

. ThenF =q on X(T) where is of finite order (2 or 3 ifW is irreducible) andthe reflection coset W is independent ofq.


Order formulaTheorem (Steinberg)

|GF|= qPn

i=1(di1)n

i=1

(qdi i)

where di is the degree of the fundamental invariants f i of W andi theeigenvalue of on fi.

The proof of Steinberg uses the Bruhat decomposition G=

wWBwB.Here we can write winstead of a representative in NG(T) since BT. IfwWF since T is connected we can choose an F-stable representative inNG(T) and we have G

F =

wWFBFwBF.

IfB=U T then BFwBF =TFUFwUw is a unique decompositionwhere UwU

F/(wUFw1).

We have |TF|= det(q ),we have |UF|= q

Pni=1(di1),

and there is a length w N(w) on WF such that|Uw|= qN(w).

Steinberg concludes by some combinatorics on W.Jean Michel (Universite Paris VII) Finite groups of Lie type Berlin, September 2009 8 / 15

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-adic cohomology

There is a nicer proof when F is a Frobenius using -adic cohomology.Choosing |q, for a variety X over Fqthere are cohomology groupsHic(X,Z) which are finite dimensional Z-modules, are 0 unless0 i2dim X, and such that:

IfXcomes from a variety X0 over a number field thendim Hic(X,Z) = dim H

ic(X0,C).

IfX = Ar is an affine space, then Hic(X,Z) = 0 unless i= 2r, andH2rc (X,Z) = Z.

We have the Lefschetz formula|XF|=

i(1)

iTrace(F |Hic(X,Z)).

Application: for any Fq-structure on the affine space X = Ar we have

|XF

|= qr; indeed F acts on H2r

c (X,Z

) by a scalar . For n large enough

the action ofFn is the standard action x xqn

son =qnr.Thus for any unipotent group |U(Fq)|= qdimU.


Cohomology ofG/B

To compute|GF| write GF =BF(G/B)F and use that

The cohomology of G/B is isomorphic to the coinvariant algebra SW.

The isomorphism multiplies the grading by 2; we have Hic(G/B,Z) = 0for iodd. It follows that|(G/B)F|= GradedTrace(F |SW).

Since SSW

SW, we have

GradedTrace(F |SW) = GradedTrace(F |S)

GradedTrace(F |SW)=

(qdii 1)

det(q 1) .

Since SSW SW, we have This works out to be the same as theformula above once multiplied by|BF|.


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Prime factors

The decomposition in prime polynomial factors of|GF| is

|GF|= qPn

i=1(di1)

d

d(q)|a(d)|

whered is the d-th cyclotomic polynomial, and a(d) ={i |did =i},

whered =e2i/d.

Here a(d) is the same set we called a(d); it does not depend on theprimitived-th root chosen since W is rational. There is a connectionwith maximal d-eigenspaces of elements ofW.

The above gives the decomposition in primes for large enough primes

=p, since gcd(d(q), d(q)) divides lcm(d, d

); thus if does notdivide |W|, it divides only one of the factors.


d-Sylows

There is a subgroup ofGF of cardinality d(q)|a(d)|.

Assume wFhas a d-eigenspace of dimension |a(d)|.Consider an F-stable torus Twcorresponding to the F-class ofw.IfTB is F-stable this is obtained as gTg1 where g1F(g) NG(T)has image w in W. Then g1 transports (Tw, F) to (T, wF), so

anF-stable subtorus ofTwcorresponds to a wF-stable subtorus ofT,or equivalently a wF-stable sublattice ofX(T).

Since wFstabilizes the lattice X(T) it has a factor d(q)|a(d)| in its

characteristic polynomial. This corresponds to a subtorus S ofTw suchthat|SF|= d(q)

|a(d)|.

Such a subtorus S is called a d-Sylow.


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Corollary of Springer-Lehrer

Theorem

Thed

-Sylows are GF-conjugate.

If S is a d-Sylow, then NGF(S)/CGF(S) is a complex reflectiongroup, with degrees{di |ia(d)}.

The number of d-Sylows is1 (mod d(q)).

Ad-Sylow is the product of|a(d)| F-stable tori Si such that SFi is a

cyclic group of orderd(q).

The third item above says that |GF|/|CGF(S)|, when specializing q tod,gives |NGF(S)/CGF(S)|.


Parabolic subgroups and Levis

A group Pcontaining a Borel subgroup is a parabolic subgroup.It has a Levi decomposition P=V L where V is its unipotent radical,and L is a reductive group; L is what is called a Levi subgroup ofG.Example: For GLn upper block-triangular matrices are the parabolicsubgroup containing the Borel of upper triangular matrices. Thecorresponding Levi is block-diagonal.

The centralizer of a torus is a Levi subgroup, thus ifS is a d-Sylow,L:= CG(S) is a Levi subgroup. We haveWG(L) =NG(L)/L= NG(S)/CG(S); this relative Weyl group is suchthat WG(L)

F is a complex reflection group. As for a torus, L is theconnected component ofNG(L); we have CG(S) =NG(S)

0.

IfScorresponds to a maximal d-eigenspace Uof some element ofWF,the Levi L= CG(S) has weyl group CW(U).


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-Sylows and d-Sylows

Assume that the-Sylow S of GF

is abelian. Then |S| divides a uniquefactord(q)

|a(d)| of GF, and there is a uniqued-Sylow S such thatSS .

It follows that

CG(S) =CG(S) is a Levi subgroup.

NGF(S)/CGF(S) =NGF(S)/CGF(S) is a complex reflection group.


michel 2

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